People in California are worried about the mathematical performance oftheir children. So worried that debates about the "right" way to teachmathematics have become so metaphorically bloody they have been termed the"math wars." Much of this concern has stemmed from students' performance onstate and national tests, but nobody seems to have stopped at any point toquestion the value of the type of knowledge assessed on these tests. Iwould like to halt the debate for a moment in order to pose this question:Is success on a short, procedural test the measure we want to adopt toassess the effectiveness of our students' learning? In other words, dothese tests assess the sort of knowledge use, critical thought, andreasoning that is needed by learners moving into the 21st century?

One of my concerns in this area is that the current debate about standardsassumes there to be one form of knowledge that is unproblematicallyassessed within tests. This is despite the fact that a largebody of research from psychological and educational fields shows theexistence of different forms of knowledge. There is also increasingevidence that students can be very successful on standard, closed testswith a knowledge that is highly inert and that they are unable to use inmore unusual and demanding situations (such as those encountered inthe workplace).

Test knowledge, in other words, is often the sort of knowledge that isnontransferable and is useful for little more than taking tests. Todemonstrate what I mean, I would like to describe the results of a researchproject that monitored the learning of students who experienced completelydifferent mathematics teaching approaches over a three-year period. Inresponse to these different approaches, the students developed differentforms of knowledge and understanding that had enormous implications fortheir effectiveness in real-world situations.

Two schools in England were the focus for this research. In one, theteachers taught mathematics using whole-class teaching and textbooks,and the students were tested frequently. The students were taught in trackedgroups, standards of discipline were high, and the students worked hard. Thesecond school was chosen because its approach to mathematics teaching wascompletely different. Students there worked on open-ended projects inheterogeneous groups, teachers used a variety of methods, and discipline wasextremely relaxed. Over a three-year period, I monitored groups of students atboth schools, from the age of 13 to age 16. I watched more than 100 lessonsat each school, interviewed the students, gave out questionnaires, conductedvarious assessments of the students' mathematical knowledge, and analyzedtheir responses to Britain's national school-leaving examination inmathematics.

At the beginning of the research period, the students at the two schoolshad experienced the same mathematical approaches and, at that time, theydemonstrated the same levels of mathematical attainment on a range of tests.There also were no differences in sex, ethnicity, or social class between thetwo groups. At the end of the three-year period, the students had developedin very different ways. One of the results of these differences was thatstudents at the second school -- what I will call the project school, asopposed to the textbook school--attained significantly higher grades onthe national exam. This was not because these students knew moremathematics, but because they had developed a different form of knowledge.

At the textbook school, the students were motivated and worked hard, theylearned all the mathematical procedures and rules they were given, and theyperformed well on short, closed tests. But various forms of evidence showedthat these students had developed an inert, procedural knowledge that theywere rarely able to use in anything other than textbook and test situations. Inapplied assessments, many were unable to perceive the relevance of themathematics they had learned and so could not make use of it. Even whenthey could see the links between their textbook work and more-applied tasks,they were unable to adapt the procedures they had learned to fit thesituations in which they were working.

The students themselves were aware of this problem, as the followingdescription by one student of her experience of the national exam shows:"Some bits I did recognize, but I didn't understand how to do them, Ididn't know how to apply the methods properly."

In real-world situations, these students were disabled in two ways. Notonly were they unable to use the math they had learned because they couldnot adapt it to fit unfamiliar situations, but they also could not see therelevance of this acquired math knowledge from school for situationsoutside the classroom. "When I'm out of here," said another student, "themath from school is nothing to do with it, to tell you the truth. Most ofthe things we've learned in school we would never use anywhere."

Students from this school reported that they could see mathematics allaround them, in the workplace and in everyday life, but they could notsee any connection between their school math and the math theyencountered in real situations. Their traditional, class-taught mathematicsinstruction had focused on formalized rules and procedures, andthis approach had not given them access to depth of mathematicalunderstanding. As a result, they believed that school mathematicalprocedures were a specialized type of school code -- useful only inclassrooms. The students thought that success in regard mathematicsto be a thinking subject. As one girl put it, "In math you have to remember;in other subjects you can think about it."

The math teaching at this textbook school was not unusual. Teachers therewere committed and hard-working, and they taught the students differentmathematical procedures in a clear and straightforward way. Their studentswere relatively capable on narrow mathematical tests, but thiscapability did not transfer to open, applied, or real-world situations. Theform of knowledge they had developed was remarkably ineffective. At theproject school, the situation was very different. And the students'significantly higher grades on the national exit exam were only a smallindication of their mathematical competence and confidence.

The project school's students and teachers were relaxed about work.Students were not introduced to any standard rules or procedures (until afew weeks before the examinations), and they did not work throughtextbooks of any kind. Despite the fact that these students were notparticularly work-oriented, however, they attained higher grades thanthe hard-working students at the textbook school on a range of differentproblems and applied assessments. At both schools, students hadsimilar grades on short written tests taken immediately after finishingwork. But students at the textbook school soon forgot what they hadlearned. The project students did not. The important difference betweenthe environments of the two schools that caused this difference inretention was not related to standards of teaching but to differentapproaches, in particular the requirement that the students at theproject-based school work on a variety of mathematical tasks and thinkfor themselves.

When I asked students at the two schools whether mathematics was more aboutthinking or memorizing, 64 percent of the textbook students chosememorizing, compared with only 35 percent of the project-based students.The students at the project school were less concerned about memorizingrules and procedures, because they knew they could think about differentsituations and adapt what they had learned to fit new and demanding problems.On the national examination, three times as many students from theheterogeneous groups in the project school as those in the trackedgroups in the textbook school attained the highest possible grade. Theproject approach was also more equitable, with girls and boys attaining thedifferent grades in equal proportions.

It would be easy to dismiss the results of this study because it wasfocused on only two schools, but the textbook school was not unusual inthe way its teachers taught mathematics. And the in-depth nature of the studymeant that it was possible to consider and isolate the reasons why studentsresponded to this approach in the way that they did. The differences in theperformance of the students at the two schools did not spring from "bad''teaching at the textbook school, but from the limitations of drawing upon onlyone teaching method. To me, it does not make any sense to set anyone particular teaching method against another and argue about which one isbest. Different teaching methods do different things. We may as well arguethat a hammer is better than a drill. Part of the success of the projectschool came from the range of different methods its teachers employed andthe different activities students worked on.

Some proponents of traditional teaching want students to follow the sametextbook method all of the time. A few students are successful in such anapproach, but the vast majority develop a limited, procedural form ofknowledge. This kind of knowledge may result in enhanced performances onsome tests, but the aim of schools must surely be to equip students with acapability and intellectual power that will transcend the boundaries of theclassroom.

-------------------------------Jo Boaler is an assistant professor of education at Stanford University inStanford, CA. Her book Experiencing School Mathematics received theOutstanding Book in Education award in the United Kingdom in 1997. She canbe reached by e-mail at <joboaler@stanford.edu>******************************************************Jerry P. BeckerDept. of Curriculum & InstructionSouthern Illinois UniversityCarbondale, IL 62901-4610 USAFax: (618)453-4244Phone: (618)453-4241 (office)E-mail: jbecker@siu.edu